enhancing one dimensional - empossible

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Computational Science:

Introduction to Finite‐Difference Time‐Domain

EnhancingOne‐Dimensional FDTD

Slide 1

Lecture Outline

•Convergence•Pure sine wave source•Grid dispersion• Incorporating loss• Frequency dependent materials•1D anisotropic FDTD

Slide 2

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Slide 3

Convergence

Convergence (1 of 2)

Slide 4

Convergence is the tendency of a numerical model to asymptotically approach a constant answer as the grid resolution (x, y, and z) and/or the time step t is reduced.

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Convergence (2 of 2)

Slide 5

Transm

ittance (d

B)

This is a better way to view and judge convergence, although it is more computationally intensive to calculate.

Conclusions on Convergence

• The equations and procedures provided in this course are only “rules of thumb” that will give you a reasonable first guess at grid resolution and time step.

• The real way to determine sufficient grid resolution and sufficiently small time step is to check for convergence.

Slide 6

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Slide 7

Pure Sine Wave Source

The Gaussian Pulse Source

Slide 8

The Gaussian pulse source is a single pulse that contains energy at all frequencies from DC up to some maximum frequency. It is not easy to make conclusions on the steady‐state response at a particular frequency by watching this pulse propagate through a device.

2

0expt

tt

g

duration of simulation

0t

0.5

B 0 6t

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The Pure Frequency Source

Slide 9

FDTD can also be excited by a “pure” sine wave of frequency f0. For a robust simulation, the amplitude of the wave should be tapered from zero to unity amplitude. Using a single frequency source eliminates many of the benefits of FDTD, but visualizing the field is more intuitive.

0

2

00

0

sin 2

exp

1

A t f t

t tt t

A t

t

t

g

t

0

0

3

3

f

t

0t

Frequency f0

Slide 10

Grid Dispersion

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The Wave Vector

Slide 11

1. Direction – The direction of the wave is perpendicular to the wave fronts.

00

2 2 free space wavelengthkn

ˆ ˆ ˆx y zk k x k y k z

2. Wavelength and Refractive Index – The magnitude of the wave vector tells us the spatial period (wavelength) of the wave inside the material. Therefore, |k| also conveys the material’s refractive index n.

The wave vector is a vector quantity that conveys two pieces of information at the same time:

Dispersion Relations

Slide 12

The dispersion relation for a material relates the wave vector to frequency. Essentially, it tells us the refractive index as a function of direction through a material.

It is derived by substituting a plane wave solution into the wave equation.

For an ordinary linear, homogeneous, and isotropic (LHI) material, the dispersion relation is:

22 2 2 2 2

0x y zk k k k nv

0 velocity of physical wavec

vn

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Index Ellipsoids

Slide 13

Isotropic Materials

k

P

x

yAnisotropic Materials

k

P

x

y

Phase propagates in the direction of k. Therefore, the refractive index derived from |k| is best described as the phase refractive index. Velocity here is the phase velocity.

Energy propagates in the direction of P which is always normal to the surface of the index ellipsoid. From this, we can define a group velocity and a group refractive index.

Index ellipsoids are a map of the refractive index as a function of direction through a material. They are constructed by plotting the dispersion relation.

Derivation of Numerical Dispersion Relation

Slide 14

We derive the numerical dispersion relation by substituting a plane wave solution into our finite‐difference equations representing Maxwell’s equations on a Yee grid.

02

0

0

02

0

0

x y z

x y z

xj fT t k I x k J y k K z

y

z

xj fT t k I x k J y k K z

y

z

E

E E e

E

H

H H e

H

1,2,3, time step

1, 2,3, grid index along x direction

1,2,3, grid index along y direction

1, 2,3, grid index along z direction

T

I

J

K

2 2

2 2

, 1, , , , , 1 , ,

, , 1 , , 1, , , ,

1, , , , , 1

,

,

,,

0

, ,

0

, , ,

, , , ,

t t

t t

i j kt tt t t t xx

i j kt tyyt

i j k i j k i j k i j kz z y y

i j k i j k i j k i j kx x z z

i j k i j k i

i j k i

j ky y x

t t t

t

j kx x

i j k

t

i j ky y

t

E E E E

E

H H

H H

y z c t

z x

E E E

E

c t

E E

x

2 2

, ,

0

,, ,

, , ,t t

ii j kx

j k i j kzi j k

tt zztz

y

HE

c t

H

2 2 2 2

2 2 2 2

, , , 1, , , , , 1

, , , , 1 , , 1, ,

, ,

, ,

0

, ,

0

, , , ,

, , , ,

t t t t

t t t t


i j

i j k i j kx x

i j k i j ky y

i j kt t t t xx t t

k i j k i j k i j kx x z z

i

t

i j kt t t t yy t t

j

t

ky

y z c t

z x

H H H H

H H H H

H

E E

c

E

t

E

2 2 2 2

1, , , , , 1 ,, ,, , , ,

0

t t t t

i j k i j k i j k i j k i jy

kz

i j kt t t tx

tz zt tx

zE E

x y c t

H H H

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Numerical Dispersion Relation

Slide 15

The numerical dispersion relation for a 3D Yee grid is

22 2 21 1 1 1

sin sin sin sin2 2 2 2

yx zk yk x k z t

x y z v t


22 21 1 1

sin sin sin2 2 2

yxk yk x t

x y v t


2 21 1

sin sin2 2zk z t

z v t

velocity of numerical wavev

Limiting Case

Slide 16

As the time step t and the grid resolution x, y, and z approach zero, the numerical dispersion relation reduces exactly to the dispersion relation of a real material.

22 2 2

22 2 2

22 2 2

1 1 1 1sin sin sin sin

2 2 2 2

2 2 2 2

yx z

yx z

x y z

k yk x k z t

x y z v t

kk k

v

k k kv

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Numerical Phase Velocity

Slide 17

The 3D numerical dispersion relation is solved for k numerically by iterating until a valid solution is found. For propagation along a single axis, the dispersion relation reduces to 1D

and can be solved analytically for 𝑘 .

pvk

The speed that a wave will travel through a grid is defined as

2 2

11 1 2sin sin sin sin

2 2 2z

z

k z t z tk

z v t z v t

This becomes

1

2

sin sin2

p

zv

z tv t

Note, waves in a grid propagate at a slightly different velocity than a physical wave due to the additional interaction with the grid.

Typically they travel slower.

Animation of Grid Dispersion

Slide 18

Simulation with strong grid dispersion...

Simulation with near‐zero grid dispersion...

Notes: • Grid dispersion error accumulates the farther a wave propagates.• Grid dispersion will be more severe on larger grids.• This is yet another consideration for grid resolution size of the grid

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Impact of the Time Step

Slide 19

The time step imposes an isotropic dispersion.

0

1

t

tN f

number of time steps per wave periodtN

Impact of the Grid Resolution

Slide 20

The grid imposes an anisotropic dispersion (spatial dispersion).

0

N

number of grid points per wavelengthN

This may seem like a small error, but it accumulates quickly on large grids.

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Methods for Mitigating Grid Dispersion

• Increase grid resolution (i.e. decrease z)•Adjust numerical phase velocity

•Use a hexagonal grid•Use higher‐order accurate derivatives•Use discrete Fourier transforms to calculate derivatives

•Adopt more advanced algorithms like M24.

Slide 21

Adjusting Numerical Phase Velocity (1 of 3)

Slide 22

The numerical phase velocity of a wave in 1D‐FDTD is

1

0

2

sin sin2

p

zv

n z tc t

0c

n The phase velocity of a wave in FDTD is different than a

physical wave due to numerical artifacts of the Yee grid.

We can force vp = c0/n by introducing a correction factor f.

0

1

0

2

sin sin2

p

c zv

n n z tcf

t

The correction factor fmultiplies the refractive index so as to adjust the phase velocity in the most straightforward manner possible.

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Slide 23

We derive an equation for the correct factor by solving this equation for f

00

0 0

sin 2

sin 2

k n zc t

n z k tf

c

In practice, the grid is not homogeneous and the refractive index is set to some average value.

0 avg0

avg 0 0

sin 2

sin 2

k n zc t

n z k tf

c

f will typically be just slightly less than 1.0.


Slide 24

Grid dispersion can be compensated by adjusting the refractive index across the grid according to the correction factor f. This is accomplished by modifying the permittivity and permeability functions as follows.

We can only perfectly compensate for dispersion at one frequency, in one refractive index, and in one direction.

In practice, we compensate at the center frequency and for some average refractive index. We also choose the dominant direction that waves travel or we can just average the correction factor over all angles or compensate at an angle of 22.5°.

r r

r r

f

f

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Example of Adjusting Numerical Phase Velocity

Slide 25

Suppose you are constructing a 1D FDTD simulation of a device in air operating at around 1.5 GHz. You have calculated your FDTD parameters to be

Compensate for numerical dispersion.

10

0

3.0 cm

50 ps

220.9585 m

z

t

fk

c

111ms00

1 11m0 0 s

sin 0.5 20.9585 m 1.0 0.03 m299792458 5 10 ssin 20.9877

sin 2 1.0 0.03 m sin 0.5 299792458 20.9585 m 5 10 s

k n zc tf

n z c k t

% COMPENSATE FOR NUMERICAL DISPERSIONf = c0*dt/(n*dz) * sin(k0*n*dz/2)/sin(c0*k0*dt/2);UR = f*UR;ER = f*ER;

Conclusions About Numerical Dispersion

•Waves on a grid propagate slower than a physical wave• The time step imposes isotropic dispersion• The grid imposes anisotropic dispersion• Primary effect is to push the spectral response to lower frequencies (longer wavelengths)•We can make the numerical dispersion arbitrarily small by increasing the grid resolution and decreasing the time step.•We can compensate for numerical dispersion by adjusting the free space permittivity and permeability

Slide 26

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Slide 27

Incorporating Loss

Maxwell’s Equations

Slide 28

An easy way to account for absorption loss in FDTD is through the material conductivity .Start with Maxwell’s equations in the following form:

0

v

DH J

t

BE

t

B

D

0 r

0 r

D t E t

B t H t

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Maxwell’s Equations with Conductivity

Slide 29

The current density 𝐽 is related to conductivity and the electric field 𝐸 through:

0 r

0 r

EH E

t

HE

t

J E

Substituting this and the constitutive relations into Maxwell’s curl equations yields

This is essentially Ohm’s law for electromagnetics.

Normalized Maxwell’s Equations

Slide 30

Just like before, the magnetic field is normalized according to:

r0

0

r

0

EH E

c t

HE

c t

0

0

H H

Substituting this into Maxwell’s curl equations leads to

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Expansion of Curl‐H Equation

Slide 31

Only one of the curl equations has changed so it is only necessary to derive new equations for the first curl equation.

r0

0

EH E

c t

This vector equation can be expanded into three scalar equations.

00

00

00

y xx xzxx x

yy yx zyy y

y x zz zzz z

H EHE

y z c t

EH HE

z x c t

H H EE

x y c t

Finite‐Difference Approximation (1 of 2)

Slide 32

Start with the first partial differential equation.

00

y xx xzxx x

H EHE

y z c t

Approximating this with finite‐differences on a Yee grid leads to:

2 2 2 2

, , , 1, , , , , 1, , , ,, ,

, , , ,

00

?

t t t t

i j k i j k i j k i j ki j k i j ki j k

z z y y i j k i j kt t t t x xxx t t txx x

H H H H E EE

y z c t

What time step should be used here?

, , , , or or ?

i j k i j k

x xt t tE E

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Finite‐Difference Approximation (2 of 2)

Slide 33

REMEMBER: All terms in a finite‐difference equation MUST exist at the same point in time and space.

Here, each term must exist at 𝑡 Δ𝑡 2⁄ , but we only have 𝐸 stored at 𝑡 or 𝑡 Δ𝑡.

𝐸 must be interpolated at 𝑡 Δ𝑡 2⁄ .

2 2 2 2

, , , 1, , , , , 1, , , , , , , ,, ,

, ,

002

t t t t

i j k i j k i j k i j ki j k i j k i j k i j ki j k

z z y y i j kt t t t x x x xxxt t t t t txx

H H H H E E E E

y z c t

2

, , , ,, ,

2t

i j k i j ki j k x xt t t

x t

E EE

The finite‐difference equation is then

Derivation of the Update Equation (1 of 3)

Slide 34

Now there is a finite‐difference equation with two occurrences of 𝐸 at the future time step. The algebra is more involved, but the equation is still just solved for 𝐸 at the future time step.

2 2 2 2

, , , 1, , , , , 1

, , , , , , , ,, ,, ,

00

2

t t t t

i j k i j k i j k i j k

z z y yt t t t

i j k i j k i j k i j ki j ki j k x x x xxxt t t t t t

xx

H H H H

y z

E E E E

c t

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Slide 35

2 2 2 2

, , , 1, , , , , 1, , , , , , , ,, ,

, ,

00

, , , , , ,, , , , , ,0 0

0

2

2 2

t t t t

i j k i j k i j k i j ki j k i j k i j k i j ki j k

z z y y i j kt t t t x x x xxxt t t t t txx

i j k i j k i j k ii j k i j k i j kxx xx xx xx

x x xt t t t t

H H H H E E E E

y z c t

E E Ec t

, ,, ,

0

, , , , , , , ,, , , ,0 0

0 02 2

j ki j k

x t

i j k i j k i j k i j ki j k i j kxx xx xx xx

x xt t t

Ec t

E Ec t c t

Expand equation and collect common terms on right side of equation.

Solve for 𝐸 at the future time step.

2 2 2 2

, , , 1, , , , , 1, , , , , , , ,

, , , ,0 0

0 02 2

t t t t

i j k i j k i j k i j ki j k i j k i j k i j k

z z y yi j k i j k t t t txx xx xx xxx xt t t

H H H HE E

c t c t y z

2 2 2 2

, , , ,

, , , 1, , , , , 10

, , , ,0, , , , , , , ,

0 0

0 0

2 1

2 2

t t t t

i j k i j k

i j k i j k i j k i j kxx xx

z z y yi j k i j k t t t t

x xi j k i j k i j k i j kt t txx xx xx xx

H H H Hc tE E

y z

c t c t


Slide 36

Rearrange the update coefficients

2 2 2 2

, , , ,

, , , 1, , , , , 10

, , , ,0, , , , , , , ,

0 0

0 0

2 1

2 2

t t t t

i j k i j k

i j k i j k i j k i j kxx xx

z z y yi j k i j k t t t t

x xi j k i j k i j k i j kt t txx xx xx xx

H H H Hc tE E

y z

c t c t

2 2 2 2

, , , 1, , , , , 1

, , , , , , , ,

1 2

, , , ,

0, , , ,

, , 001 , , , ,

0 0

0

22

2

2

t t t t


z z y yi j k i j k i j k i j k t t t t

x Ex x Ext t t

i j k i j k

xx xxi j k i j k

i j k xx xxEx i j k i j k i

xx xx xx

H H H HE m E m

y z

tc tm

c t

, , , ,

, , 0 02 , , , , , , , ,

0 0

0

21

2

2

j k i j k

xx

i j k

Ex i j k i j k i j k i j k

xx xx xx xx

t

c tm

t

c t

The update coefficients have gotten big and ugly!

These will get even worse later.

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Confirmation We Are Probably Correct

Slide 37

A good test to see if there is a mistake is to set the conductivity to zero and see if the update equation reduces to the standard update equation.

2 2 2 2

, , , 1, , , , , 1

, , , , , , , ,

1 2

, , , ,

0, ,

1

2

t t t t


z z y yi j k i j k i j k i j k t t t t

x Ex x Ext t t

i j k i j k

xx xxi j k

Ex

H H H HE m E m

y z

tm

, , , ,

02i j k i j k

xx xxt

, , 0 02 , , , ,

0

1

2

2

i j k

Ex i j k i j k

xx xx

c tm

t

0

, ,i j k

xx

c t

Slide 38

Frequency Dependent Materials

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What are Frequency‐Dependent Materials

Slide 39

The electromagnetic properties of all materials change with frequency due to the physics of the mechanisms producing electric and magnetic responses.

r r

r r

t

t

Frequency‐Domain Maxwell’s Equations Time‐Domain Maxwell’s Equations

0 r

0 r

H j D

E j B

D E

B H

0 r

0 r

D tH t

t

B tE t

t

D t t E t

B t H t

Generalized Flow of Maxwell’s Equations

Slide 40

As more complicated properties of materials are incorporated into an FDTD simulation, it makes more sense to generalize the flow of Maxwell’s equations. This “compartmentalizes” the materials problem into a single and much simpler equation.

Our Current Model

Generalized Model

We will now need four sets of update equations!

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Lorentz Model for Dielectrics

Slide 41

Equation of Motion

Lorentz Model

We can use the equation of motion to model the motion of electrons bound to an atom.

2p

r 2 20

22p

0 e

1j

Nq

m

Typical Lorentz Response

Slide 42

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Drude Model for Metals

Slide 43

Drude Model

2p

r 20

1

2p

22

22p

0 e

1jj

Nq

m

The electrons in metals are not bound to a nucleus so there is no restoring force and hence no resonant frequency. The Lorentz model reduces to the Drude model when 𝜔 0.

Typical Drude Response

Slide 44

2 2Re 0 1r p

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Real Materials Have Multiple Resonances

Slide 45

At a macroscopic level, all resonance mechanisms can be characterized using the Lorentz model. This allows any number of resonances to be accounted for through a simple summation.

2r p 2 2

1 0,

1M

m

m m m

f

j

th

th0,

th

Number of resonators

Oscillator strength of the resonator

Natural frequency of the resonator

Damping rate of the resonato

m

m

m

M

f m

m

m r

r

Overall trend of decreasing dielectric constant.

Dielectric constant almost always increasing with frequency.

The Dielectric Constitutive Relation

Slide 46

𝐷 and 𝐸 are related through the constitutive relation

0 rD E

This can also be written in terms of the material polarization 𝑃.

0D E P

The material polarization 𝑃 can be written in terms of a sum of the generalized Lorentz‐Drude model.

2p 2 2

1 0,

Mm

m m m

fP E

j

In practice, as few oscillators M as possible are used that still gives accurate material properties in the frequency range of interest.

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Conversion to the Time‐Domain

Slide 47

The polarization due a single Lorentz oscillator is

2p

2 20,

mm

m m

fP E

j

Multiplying both sides by the denominator on the right leads to

2 2 20, pm m m m m mP P j P f E

This equation is then converted to the time domain

22 20, pm m m m m mP j P j P f E

2

2 20, p2m m m m m mP t P t P t f E t

t t

2

2 20, p2 m m m m m mP t P t P t f E t

t t

Governing Equations

Slide 48

It is best to not work with a second‐order time derivatives so a polarization current 𝐽 (called displacement current in EM texts) is defined.

m mJ t P tt

Now there is a system of two equations (governing equations) that only contain first‐order time derivatives.

2 20, pm m m m m m

m m

J t J t P t f E tt

J t P tt

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Update Equations for 𝑃

Slide 49

Update equations can now be derived for 𝐽 and 𝑃 that will need to be updated at every point in the grid where that material exists.

Observe that 𝐽 and 𝑃 are staggered in time just like was done for 𝐸 and 𝐻. It is 𝑃 that is

directly combined with 𝐸, so let 𝑃 exist at the integer time steps and 𝐽 exist at the half‐integer time steps.

, , , , , ,

, , ,2

, , , , , ,

, , ,2

, , , , , ,

, , ,2

i j k i j k i j k

tx m x m x mt t t t

i j k i j k i j k

tm m y m y m y mt t t t

i j k i j k i j k

tz m z m z mt t t t

P P t J

J t P t P P t Jt

P P t J

Update Equations for 𝐽

Slide 50

Start with the governing equation.

2 20, pm m m m m mJ t J t P t f E t

t

Approximate with finite‐differences

, , , , , , , ,

, , , ,, , , ,, , 22 2 2 2

0, ,

, , , ,, , 2p ,

2


t t t tx m x m x m x mt t t t i j k i j ki j k

m m x m t

i j k i j ki j k

m x m t

J J J JP

t

f E

Solve for the future value of 𝐽.

, ,, , , ,20, , ,, , , , , , , ,2

, , , p ,, , , , , ,2 2

22 2

2 2 2

i j ki j k i j kmm m i j ki j k i j k i j k i j k

t tx m x m x m x mi j k i j k i j kt t t tm m m

tt f tJ J P E

t t t

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Update Equations for 𝐸

Slide 51

The constitutive relation for multiple resonances was

01

M

mm

D t E t P t

Approximating this numerically leads to

, , , ,, , , ,

, 0 ,1

Mi j k i j ki j k i j k

x m x x mtt tm

D E P

Solving this for 𝐸 at the future time step gives the update equation.

, , , ,, ,

, ,, ,10

1 Mi j k i j ki j k

x x m x mt i j k t tm

E D P

Revised Block Diagram of Main FDTD Loop for Dispersive Dielectric Materials

Slide 52

Update 𝐻 from 𝐸

Handle 𝐻 TF/SF source

Update 𝐽 from 𝑃 and 𝐸

Update 𝑃 from 𝐽

Update 𝐷 from 𝐻

Handle 𝐷 TF/SF source

Update 𝐸 from 𝐷 and 𝑃

Loop over time

Note: We could do the same with B=H to handle dispersive magnetic materials.

, , , , , , , ,, , , , , ,

, 1 , 2 , 3 ,2 2

i j k i j k i j k i j ki j k i j k i j kt tx m Jx x m Jx x m Jx x mt t t t

J m J m P m E

, , , , , ,

, , ,2

i j k i j k i j k

tx m x m x mt t t tP P t J

, , , ,, ,

, ,, ,10

1 Mi j k i j ki j k

x x m x mt i j k t tm

E D P

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Slide 53

1D Anisotropic FDTD

Starting Point

Slide 54

BE

t

DH J

t

Start with Maxwell’s equations in the following form

0

vD

B

and the constitutive relations for anisotropic media.

D E

B H

Assuming no sources (i.e. 𝜌 0 and 𝐽 0), the equations above becomeB

Et

DH

t

0

0

D

B

D E

B H

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Normalize the Fields

Slide 55

Normalize the fields according to

0

1E E

0

0

1

1

BE

c t

DH

c t

r

r

D E

B H

0

0

D

B

0

1B B

0D c D

The governing equations become

Expansion into Cartesian Coordinates

Slide 56

Expand Maxwell’s curl equations to get

and the constitutive relations expand to

x xx x xy y xz z

y yx x yy y yz z

z zx x zy y zz z

D E E E

D E E E

D E E E

0

0 0

0

1

1 1

1

y xz

yx z

y x z

E BE

y z c t

BE EBE

c t z x c t

E E B

x y c t

0

0 0

0

1

1 1

1

y xz

yx z

y x z

H DH

y z c t

DH HDH

c t z x c t

H H D

x y c t

x xx x xy y xz z

y yx x yy y yz z

z zx x zy y zz z

B H H H

B H H H

B H H H

These equations are not easily solved for 𝐸 and 𝐻!!

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Constitutive Relations

Slide 57

The constitutive relations must be solved for 𝐸 and 𝐻.

This is most easily accomplished using the impermittivity 𝜓 and impermeability 𝜉tensors.

x xx x xy y xz z

y yx x yy y yz z

z zx x zy y zz z

E D D D

E D D D

E D D D

x xx x xy y xz z

y yx x yy y yz z

z zx x zy y zz z

H B B B

H B B B

H B B B

r 1

r

E D

D E

r 1

r

H B

B H

These new equations expand to

Calculating the Impermittivity Tensor 𝜓

Slide 58

The impermittivity tensor 𝜓 cannot be calculated by just inverting the individual tensor elements of the permittivity. That is

1ij

ij

1

xx xy xz xx xy xz

yx yy yz yx yy yz

zx zy zz zx zy zz

Instead, the entire impermittivity tensor 𝜓 is calculated by inverting the entire permittivity tensor 𝜀 .

% CALCULATE IMPERMITTIVITY TENSOR ON 2X GRIDZ = zeros(Nx2,Ny2,Nz2);Y2xx = Z; Y2xy = Z; Y2xz = Z;Y2yx = Z; Y2yy = Z; Y2yz = Z;Y2zx = Z; Y2zy = Z; Y2zz = Z;for m = 1 : Nx2*Ny2*Nz2

er = [ ER2xx(m) ER2xy(m) ER2xz(m) ; ...ER2yx(m) ER2yy(m) ER2yz(m) ; ...ER2zx(m) ER2zy(m) ER2zz(m) ];

y = inv(er);Y2xx(m) = y(1,1); Y2xy(m) = y(1,2); Y2xz(m) = y(1,3);Y2yx(m) = y(2,1); Y2yy(m) = y(2,2); Y2yz(m) = y(2,3);Y2zx(m) = y(3,1); Y2zy(m) = y(3,2); Y2zz(m) = y(3,3);

end

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Finite‐Difference Approximation of Curl Equations

Slide 59

0

0

0

0

0

0

1

1

1

1

1

1

y xz

yx z

y x z

y xz

yx z

y x z

E BE

y z c t

BE E

z x c t

E E B

x y c t

H DH

y z c t

DH H

z x c t

H H D

x y c t

2 2

2 2

, 1, , , , , 1 , ,

, , 1

0

, , 1, , , ,

1, , , , , 1

0

,

, , , ,

, , , ,

1

1

t t

t t


i j k i j k

i j k i j kx x

i j k i j ky

i j k i j kx x z z

i j k i j

t tt

k i j ky y

t t t

t tt t t t

t t x t

y

E E E E

E E E E

E E

B B

y z c t

z x c t

x

E

B B

E

2 2

2 2 2 2

2 2 2 2

0

0

, , , ,

, , , 1, , , , , 1

, , , , 1 , , 1, ,

, ,

, , , ,

0

, ,

1

1

1

t t

t t t t

t t t t

t tt

t t t

i j kx

i j k i j kx x

i j

i j k i j kz z


i j k i j k i j k

t t t t

t

i j kx x z zt t t y

B B

H H H H

H H H H

y c t

y z

D D

c t

z x

D

c

2 2 2 2

, , 1, , , ,

0

, ,

, ,, 1 ,, ,1t t t t

i j k i j k i j k i j ky y x x

t t t

k i j ky

i j k i j kz zt t t t t t t

D

D D

t

x y c t

H H H H

Finite‐Difference Approximation of Constitutive Relations (1 of 2)

Slide 60

1

, ,

1,

,

1

,

, , 1, 1, , , ,

1, , 1, , 1 , , , , 1

, , , ,

1, 1, , 1, , , , ,

1, , 1, , 1 , , , 1

4

4

i j kxx

i j k i j k i j k i j kxy xy xy xy

i j k

i j k i j kx x

i j k i j k i j k i j ky y y y

i j k i j k i j k ii jz

k i j k i j kxz xz xz x

jz

kz z z

iy

E D

D D D D

D D D D

E

1, , 1, , 1, , , 1, 1, ,

,

, , , 1, 1, 1,,

, ,

, , , , 1 , 1, , 1, 1

, ,

,

, , , , 1 , 1, , 1, 1

4

4

i j k i j k i j k i j kx x x xj k

i

i j k i j k i j k i j kyx yx yx yx

i j kyy

i j k i j k i

j ky

i j k i j k i j k iz

j kz z z z

k

j k i j ky yz y

i

y

z

z

j

z

D D D D

D

D D D D

E

, , , , 1 1, , 1, , 1

, , , , 1 , 1

, , , , 1 1, , 1, , 1

, , , , 1 , 1, , 1, 1, , 1, 1

, , , ,

4

4

i j k i j k i j k i j kz

k

x zx zx zx

i j k i j k

k

i j k i j kzy zy zy zy

i j

i j i j k i j k i j kx x x x

i j k i j k i j

z

k i j ky y y y

ik jz z

D D D D

D D D D

D

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Finite‐Difference Approximation of Constitutive Relations (2 of 2)

Slide 61

, , , ,

, , 1, , , 1, 1, 1,

, , 1, , , , 1 1, , 1

, ,

,

, ,

, , 1, , , 1, 1, 1,

, , 1, , , , 1 1 , 1

4

4

i j k i j kx x

i j k i j k i j k i j ky y y y

i j k i j k

i j kxx

i j k i j k i j k i j kxy xy xy xy

i j k i j k i j

y

k i j kxz

i j k i j kz z z zxxz z

i

xz

j

H B

B B B B

B B B B

H

,, , 1, ,, , 1, , , 1, 1, 1

, ,

, , , 1,

1

, , 1

, , 1, 1,

, ,

, ,, , 1, , , 1 , 1, 11

,

1

,

,

4

4

i j k i j k i j k i j kx x x xk

i

i j k i j k i j k i j kyx yx yx yx

i j kyy

i j k

i j k

i j k i j k i j ky

i

y

i j k i j k i jyz yz yz z

zx

k i j kz z z z

j kz

B B B B

B

B B B B

H

, , 1, , , , 1 1, , 1

,

, , 1, , , , 1 1, , 1

, , , 1, , , 1 , 1, 1, , 1, , , 1 , 1, 1

, , , ,

4

4

i j k i j k i j k i j kx x x x

i j k i j k i j k i j ky

k

j k i j k i j k i j kzx zx zx

i j k i j k i j k i j kzy zy zy zy

i j kz

y y y

i jzz

B B B B

B B B B

B

Reduction to 1D (1 of 2)

Slide 62

0x y

2 2

2 2

2 2

2 2

1

1

0

0

1

0

0

1

1

1

0

0

1

1

t t

t t

t t

t t

t tt t

t t

k kx

k ky y

k kx x

k kx x

k ky y

z

x

k ky y

z

k ky y

k

t t

t t t t t

t t t t t

kx x

z c t

z c t

z c t

z c t

E E

E

B B

B BE

D D

D D

D

B

H H

H H

For 1D problems,

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Reduction to 1D (2 of 2)

Slide 63

11 11

2 2

k k kx x y

k k ky x y

k kk ky yk

k kxx xy

k kyx yy

k

x

kk kzy zyzx zx x

z

E D D

E D D

D DD DE

111 1

2 2

k k kx x y

k k ky x y

k kk ky yk x

k kxx xy

k kyx yy

k kk kzy zyzx x xz

z

H B B

H B B

B BB BH

Update Equations (1 of 2)

Slide 64

2 2

2 2

2 2

2 2

1

1

1

1

t t

t t

t t

t t

k kx x

k

t t t t

t t t t

t tt t t

t t

k ky y

k kx x

k kx x

k

ky y

k

ky

ky y

k kx xtyt t

E E

E E

D

m

m

m

B B

B

D

Hm

B

H

HD D

H

0t cm

z

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Update Equations (2 of 2)

Slide 65

11 11

2 2

k k kx x y

k k ky x y

k kk ky yk

k kxx xy

k kyx yy

k

x

kk kzy zyzx zx x

z

E D D

E D D

D DD DE

111 1

2 2

k k kx x y

k k ky x y

k kk ky yk x

k kxx xy

k kyx yy

k kk kzy zyzx x xz

z

H B B

H B B

B BB BH

TF/SF Source (1 of 2)

Slide 66

The following update equations span the TF/SF interface. In the SF region, these are

the 𝐵 field update equations.

2 2

1t tt t

k kx x

ky t

ky t

B E EmB

2 2

1t tt t

k ky y

kx t

kx t

B E EmB

2 2

1t t

k ky t tt t t

k kx x ym H HD D

2 2

1t t

k kx t tt t t

k ky y xm H HD D

In the TF region, these are the 𝐷 field update equations.

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TF/SF Source (2 of 2)

Slide 67

The 𝐵 field update equations exist in the SF region but contain TF terms. It is necessary to subtract the source from the TF terms.

src src

src s

ssr

r

src rc

rc s

2

r s

2

c

c

2

c

1,src

,s1

c

1

r

1

1

t t

t

t t t t t

t t t t

k ky y

k

k kx x

k

y

ky

ky

k

yx E

B

m

E Em

EB E

E

m

B

src src

src s

ssr

r

src rc

rc s

2

r s

2

c

c

2

c

1,src

,s1

c

1

r

1

1

t t

t

t t t t t

t t t t

k kx x

k

k ky y

k

x

kx

kx

k

xy E

B

m

E Em

EB E

E

m

B

s

src s

2

rc

r

srcr

sc

2 2 2

2

s

s c sr

r rr c

c

2

c s c

1 1

11

t t t

t t t

ky

ky

k kx

k ky y

k ky y

x t tt t t t

t tt t

kx

m

m m

HD D

D

H

H H

H

H

s

src s

2

rc

r

srcr

sc

2 2 2

2

s

s c sr

r rr c

c

2

c s c

1 1

11

t t t

t t t

kx

kx

k ky

k kx x

k kx x

y t tt t t t

t tt t

ky

m

m m

HD D

D

H

H H

H

H

The 𝐷 field update equations exist in the TF region but contain SF terms. We must add the source to the SF terms.

The Source Terms

Slide 68

src

src

src

2

src

2

src

src

1 ,incsrc inc

,inc 0

1 ,incsrc inc

,inc 0

2 2

2 2

t

t

k

x xt

k

y yt

k rx y x xt

r

k ry x y yt

r

E P g t

E P g t

n z tH P g t

c

n z tH P g t

c

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Sequence of Update Equations

Slide 69

Step 1 ‐‐ Update 𝐵

2 2 2 2

1 1 t t t tt t t t t t

k k k kx t

k k k ky y x xx y tyB B BE E Em m EB

2 2 2 2

1 1 t t t tt t t tt t t t t t

k k k ky y

k k k kx y xy xxD D D DH H Hm Hm

Step 2 – Update 𝐻11

, , ,11

, , , 2

2

k kk kzy zyk k k k zx z

k kk ky yi j k k k i j k k k i j k x x

x x y y x y zx

xx xy yx yy

B BB BH B B H B B H

Step 3 – Update 𝐷

Step 4 – Update 𝐸

11 11

2 2

k k kx x y

k k ky x y

k kk ky yk

k kxx xy

k kyx yy

k

x

kk kzy zyzx zx x

z

E D D

E D D

D DD DE

Block Diagram of Main FDTD Loop

Slide 70

Update 𝐵

Correct 𝐵 for TF/SF

Update 𝐻

Record 𝐻 at z‐Low Boundary

Update 𝐷

Correct 𝐷 for TF/SF

Update 𝐸

Record 𝐸 at z‐High Boundary

Main FDTD Loop

z

z

z

z

69

70

enhancing one dimensional - empossible

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